Neutrino-nucleus cross sections relevant to detection of supernova and reactor neutrinos as well as synthesis of elements in supernova explosions are evaluated by shell-model calculations with new shell-model Hamiltonians, which can describe spin degree’s of freedom in nuclei very well. The new Hamiltonians, SFO for p-shell, GXPF1J for pf-shell and VMU (monopole-based universal interaction), have proper tensor components in common, which lead to proper shell-evolutions. We can now reproduce experimental ν-induced cross sections available, that is, those of 12C and 56Fe, and evaluate the cross sections with sound reliability in various light and medium heavy nuclei. Updated ν-induced cross sections on 12C, 13C, 16O, 56Fe, 56Ni and 40Ar are presented and compared to those previously obtained. The cross sections on 12C are applied to study light element synthesis in supernova explosions and neutrino oscillation parameters. Coherent scattering cross sections on 12C and 13C can be good probes of neutron distributions of the nuclei. 13C is an attractive target for the detection of very low energy neutrinos below 10 MeV. Accurate evaluations of ν-induced cross sections are carried out for 16O and 40Ar, which are powerful targets for the detection of supernova neutrinos. Cross sections folded over neutrino spectra with and without the oscillation effects are also compared to each other to see how they are sensitive to the oscillation effects.